1. Field of the Invention
The present invention relates to phase interferograms, and in particular to global methods to unwrap wrapped phases of an interferogram.
2. Description Of Related Art
In the field of interferometric synthetic aperture radar (IFSAR) interferograms are produced. For example, two SAR images are registered pixel by pixel and interferometrically combined to form an interferogram having precise phase differences at each pixel. These interferograms are used to generate interferometric (IF) synthetic aperture radar (SAR) products such as terrain elevations.
A known method to produce an interferogram is to multiply, pixel by pixel, a master image by the complex conjugate of a slave image. Then, on a pixel by pixel basis, recover the wrapped phase angle by calculating the inverse tangent of the ratio of imaginary component divided by the real component.
There are two approaches to phase unwrapping. The first consists of seeking to build paths from pixel to pixel across the interferogram. Phase transitions across a 2.pi. boundary can be detected and then 2.pi. may be either added or subtracted to form an unwrapped phase along the selected path.
The other approach is global and is referred to as the least squares method or Green's function method. If the relations governing the unwrapped phase are written as a system of difference equations, the equations will have the same form as the finite element equations associated with the solution to Poisson's partial differential equation in two dimensions with Neumann boundary conditions. Known computational methods are very time consuming and still do not obtain accurate results in "bad spots" (e.g., when the present invention is used in IFSAR, a bad spot might be a large body of water where the flatness of the water in the master image is different than the flatness in the slave image).
Synthetic aperture radar (hereinafter SAR) is a coherent, microwave imaging system with day, night and all-weather capability. SAR data acquisition is based on exploitation of Doppler signatures. The along track motion of a radar platform combined with a relatively large antenna beamwidth results in the recording of a large number of radar returns from each scatterer on the ground as it passes through the antenna aperture or pattern, thus forming a "synthetic aperture" along the vehicle trajectory with a length determined by the along track separation of the points at which a given target is acquired and then lost. For instance, the synthetic aperture for SEASAT was over 18 kilometers long, comprising over 4,096 individual returns from each individual scatterer. These data are collected by coherent (i.e., in-phase and quadrature-phase) demodulation to preserve their relative phases and then are processed into an image by individually adjusting their phases and adding them coherently in a fashion mathematically similar to that used to focus data from an array antenna. See D. A. Ausherman et al., "Developments In Radar Imaging," IEEE Trans. Aerosp. Electron. Syst., Volume AES-20, No. 4, pp. 363-399, July, 1984, incorporated herein by reference, for a description of SAR processing. The result of this processing is an image whose pixels are complex quantities. The phases of these individual pixels depend on: (1) the particular phase response of the individual scatterer and (2) the range to the scatterer.
A variety of space based radars have been deployed in satellites to generate information for processing as a synthetic aperture radar (e.g., SEASAT, ERS-1, ERS-2 and RADARSAT). The satellite typically orbits about 300 nautical miles above the earth in a polar orbit (i.e., passing over the north and south poles) or near polar orbit while its radar is trained at a fixed angle with respect to its NADIR (i.e., down). As the satellite orbits the earth, the radar's antenna pattern causes an illuminated spot to sweep across the earth's surface in a north-south direction at a rate defined by the satellite's velocity.
FIGS. 35 and 36 depict plan and elevation views of this illumination geometry, respectively. In FIG. 36, the satellite may orbit at, for example 800 kilometers altitude, and train the central axis of the antenna beam at a beam position with respect to NADIR, for example 20.5 degrees (SEASAT). The elevation beamwidth of the antenna pattern is, for example 6.2 degrees, resulting in an illumination area between a minimum range and a maximum range of about 100 kilometers. In FIG. 35, the antenna pattern is trained perpendicular to the velocity of the satellite, thus sweeping out a swath across the earth's surface from the minimum range to the maximum range, for example 100 kilometers. The beamwidth in the azimuth direction of the radar is, for example 1.1 degrees, thus causing the illumination of the earth in the along track direction to be about 18 kilometers (in this example).
FIG. 37 is a simplified diagram of the SEASAT system. The onboard satellite includes a stalo (i.e., stable local oscillator) and a linear FM modulator that provides a swept FM signal to the transmitter and the combiner. The transmitter forms a gated pulse from the linear FM modulator output to be passed through a circulator to the antenna for transmission. The gate radiated pulse is reflected from the earth (e.g., FIGS. 100-102) and received in the antenna where it is passed through the circulator to the receiver. The receiver advantageously alters the gain as a function of time since the transmission pulse in a process known as sensitivity time control. The output of the receiver is passed to the combiner where it is combined with the output of the linear FM modulator, and the combined signal are transmitted through a data link to a ground station. At the ground station the data link is received synchronously, demodulated and passed to a radar data recorder and formatter. In the SEASAT system, the radar data recorder and formatter converts the signals into digital form and records the data on a high-density magnetic tape recorder. Magnetic tapes from this tape recorder are then processed by radar processors in various ways defined by the user.
The linear FM modulation in the SEASAT system is used to produce accurate range data. In FIG. 38, there is depicted the output of the linear FM modulator as a solid line. As depicted in FIG. 38, the frequency of this output signal increases in a sawtooth fashion at a slope defined by (.delta.F)/(.delta.T). The transmitter amplifies the signal and gates it into a transmit pulse as depicted. The transmit pulse is radiated and after a period of time (i.e., .DELTA.T) an echo of the transmit pulse is received as a received pulse. By this time the frequency of the signal from the linear FM modulator has increased whereas the frequency of the received pulse is approximately the same as it was when it was transmitted. Within the ground station, or a subsequent process, it is possible to measure the frequency difference in the linear FM Modulator transmit and receive pulses. Since the rate of change of frequency with respect to time is known, this frequency difference is used to accurately measure the quantity .DELTA.T. From this, it is easy to determine the range of the object causing the reflection. To define a plurality of range cells, the radar processor (not shown) samples the received pulse at a plurality of points during the received pulse. These samples are then Fourier processed to determine the spectrum. The spectrum (i.e., frequencies) of the return pulse defines the various range cells.
FIG. 39 depicts an antenna pattern from a single pulse. The center line the antenna pattern is directed perpendicular to the velocity so that forward of this center line, the return signal will express plus Doppler and aft of this center line, the return signal will express minus Doppler. The range cells are determined as previously described, for example a single range cell depicted as range cell N.
FIG. 40 depicts the antenna pattern from two successive pulses, a first pulse and a second pulse. Within range cell N, there is depicted two points: P1 and P2. P1 is found within the envelope of the antenna pattern from both the first pulse and the second pulse. However, during the first pulse, P1 expresses a plus Doppler since it is forward of the center line of the antenna pattern, and during the second pulse P1 expresses a minus Doppler since it is aft of the center line of the antenna pattern. FIG. 41 depicts the Doppler expressed by a reflection from P1 (i.e., the P1 curve). Marked on this curve is the location of the frequency response expressed from the first pulse (i.e., plus Doppler) and the frequency response expressed from the second pulse (i.e., minus Doppler). Additional pulses transmitted and received by the radar win fin out the entirety of the P1 curve.
In FIG. 40, P2 is depicted as within the antenna pattern of only the second pulse, and having a plus Doppler expressed. In FIG. 42, there is depicted the Doppler curves for both PI and P2. At the point and time of the second pulse, the P1 curve indicates expression of a minus Doppler and the P2 curve indicates an expression of a plus Doppler. It will be appreciated that other points within range cell N (FIG. 40) will fill out a family of Doppler curves in FIG. 42. It is also apparent that the second pulse expresses a different Doppler frequency for point P1 than it does for point P2, and that measuring this difference in frequency, and with a certain amount of additional radar processing, the separation between points P1 and P2 within range cell N can be determined.
FIG. 43 depicts such radar processing. In the upper left hand corner, range cells from each pulse are successively stored in a corner turn memory. A complete radar image may be processed from as many as, for example, four thousand pulses. Next, a data vector is extracted from the corner turn memory representing a single range cell from each of a series of these successive pulses. This vector is Fourier processed to form an output vector representative of the spectrum of reflections from a single range cell. This spectrum represents azimuth position within the antenna pattern of the various scatterers (e.g., points P1 and P2 in FIG. 40). The vector processed from a single range cell is then collected into an output array. The processed vector from all range cells representing a two dimensional image of range by Doppler. The Doppler dimension corresponds to the azimuth position. It will be appreciated by persons skilled in the art that various processing methods are available to correct the output image into rectilinear coordinates and correct for depth of focus. When processing is carried out as complex quantities, phase information is preserved. It is this output array, and in particular the phase information in the complex quantities in this output array, that provides data for many applications such as interferometry with the synthetic aperture radar image data.